d01fc attempts to evaluate a multidimensional integral (up to dimensions), with constant and finite limits, to a specified relative accuracy, using an adaptive subdivision strategy.
Public Shared Sub d01fc ( _ ndim As Integer, _ a As Double(), _ b As Double(), _ ByRef minpts As Integer, _ maxpts As Integer, _ functn As D01..::..D01FC_FUNCTN, _ eps As Double, _ <OutAttribute> ByRef acc As Double, _ <OutAttribute> ByRef finval As Double, _ <OutAttribute> ByRef ifail As Integer _ )
public: static void d01fc( int ndim, array<double>^ a, array<double>^ b, int% minpts, int maxpts, D01..::..D01FC_FUNCTN^ functn, double eps, [OutAttribute] double% acc, [OutAttribute] double% finval, [OutAttribute] int% ifail )
- Type: System..::..Int32On entry: , the number of dimensions of the integral.Constraint: .
- Type: array<System..::..Double>()An array of size [ndim]On entry: the lower limits of integration, , for .
- Type: array<System..::..Double>()An array of size [ndim]On entry: the upper limits of integration, , for .
- Type: System..::..Int32%On entry: must be set to the minimum number of integrand evaluations to be allowed.On exit: contains the actual number of integrand evaluations used by d01fc.
- Type: System..::..Int32On entry: the maximum number of integrand evaluations to be allowed.Constraints:
- , where .
- Type: NagLibrary..::..D01..::..D01FC_FUNCTNfunctn must return the value of the integrand at a given point.
A delegate of type D01FC_FUNCTN.
- Type: System..::..DoubleOn entry: the relative error acceptable to you. When the solution is zero or very small relative accuracy may not be achievable but you may still set eps to a reasonable value and check for the error exit .Constraint: .
- Type: System..::..Double%On exit: the best estimate obtained for the integral.
d01fc returns an estimate of a multidimensional integral over a hyper-rectangle (i.e., with constant limits), and also an estimate of the relative error. You set the relative accuracy required, return values for the integrand via a method argument functn, and also set the minimum and maximum acceptable number of calls to functn (in minpts and maxpts).
The method operates by repeated subdivision of the hyper-rectangular region into smaller hyper-rectangles. In each subregion, the integral is estimated using a seventh-degree rule, and an error estimate is obtained by comparison with a fifth-degree rule which uses a subset of the same points. The fourth differences of the integrand along each coordinate axis are evaluated, and the subregion is marked for possible future subdivision in half along that coordinate axis which has the largest absolute fourth difference.
If the estimated errors, totalled over the subregions, exceed the requested relative error (or if fewer than minpts calls to functn have been made), further subdivision is necessary, and is performed on the subregion with the largest estimated error, that subregion being halved along the appropriate coordinate axis.
The method will fail if the requested relative error level has not been attained by the time maxpts calls to functn have been made; or, if the amount lenwrk of working storage is insufficient. A formula for the recommended value of lenwrk is given in [Parameters]. If a smaller value is used, and is exhausted in the course of execution, the method switches to a less efficient mode of operation; only if this mode also breaks down is insufficient storage reported.
Genz A C and Malik A A (1980) An adaptive algorithm for numerical integration over an N-dimensional rectangular region J. Comput. Appl. Math. 6 295–302
van Dooren P and de Ridder L (1976) An adaptive algorithm for numerical integration over an N-dimensional cube J. Comput. Appl. Math. 2 207–217
Note: d01fc may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the method:
On entry, , or , or maxpts is too small, or , or .
A relative error estimate is output through the parameter acc.
This example estimates the integral
The accuracy requested is one part in .